MAYBE 210.12/60.04 MAYBE 210.12/60.04 210.12/60.04 We are left with following problem, upon which TcT provides the 210.12/60.04 certificate MAYBE. 210.12/60.04 210.12/60.04 Strict Trs: 210.12/60.04 { f(x1) -> n(c(c(x1))) 210.12/60.04 , n(f(x1)) -> f(n(x1)) 210.12/60.04 , n(s(x1)) -> f(s(s(x1))) 210.12/60.04 , c(f(x1)) -> f(c(c(x1))) 210.12/60.04 , c(c(x1)) -> c(x1) } 210.12/60.04 Obligation: 210.12/60.04 derivational complexity 210.12/60.04 Answer: 210.12/60.04 MAYBE 210.12/60.04 210.12/60.04 None of the processors succeeded. 210.12/60.04 210.12/60.04 Details of failed attempt(s): 210.12/60.04 ----------------------------- 210.12/60.04 1) 'Fastest (timeout of 60 seconds)' failed due to the following 210.12/60.04 reason: 210.12/60.04 210.12/60.04 Computation stopped due to timeout after 60.0 seconds. 210.12/60.04 210.12/60.04 2) 'Inspecting Problem... (timeout of 297 seconds)' failed due to 210.12/60.04 the following reason: 210.12/60.04 210.12/60.04 The weightgap principle applies (using the following nonconstant 210.12/60.04 growth matrix-interpretation) 210.12/60.04 210.12/60.04 TcT has computed the following triangular matrix interpretation. 210.12/60.04 Note that the diagonal of the component-wise maxima of 210.12/60.04 interpretation-entries contains no more than 1 non-zero entries. 210.12/60.04 210.12/60.04 [f](x1) = [1] x1 + [1] 210.12/60.04 210.12/60.04 [n](x1) = [1] x1 + [0] 210.12/60.04 210.12/60.04 [c](x1) = [1] x1 + [0] 210.12/60.04 210.12/60.04 [s](x1) = [1] x1 + [0] 210.12/60.04 210.12/60.04 The order satisfies the following ordering constraints: 210.12/60.04 210.12/60.04 [f(x1)] = [1] x1 + [1] 210.12/60.04 > [1] x1 + [0] 210.12/60.04 = [n(c(c(x1)))] 210.12/60.04 210.12/60.04 [n(f(x1))] = [1] x1 + [1] 210.12/60.04 >= [1] x1 + [1] 210.12/60.04 = [f(n(x1))] 210.12/60.04 210.12/60.04 [n(s(x1))] = [1] x1 + [0] 210.12/60.04 ? [1] x1 + [1] 210.12/60.04 = [f(s(s(x1)))] 210.12/60.04 210.12/60.04 [c(f(x1))] = [1] x1 + [1] 210.12/60.04 >= [1] x1 + [1] 210.12/60.04 = [f(c(c(x1)))] 210.12/60.04 210.12/60.04 [c(c(x1))] = [1] x1 + [0] 210.12/60.04 >= [1] x1 + [0] 210.12/60.04 = [c(x1)] 210.12/60.04 210.12/60.04 210.12/60.04 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 210.12/60.04 210.12/60.04 We are left with following problem, upon which TcT provides the 210.12/60.04 certificate MAYBE. 210.12/60.04 210.12/60.04 Strict Trs: 210.12/60.04 { n(f(x1)) -> f(n(x1)) 210.12/60.04 , n(s(x1)) -> f(s(s(x1))) 210.12/60.04 , c(f(x1)) -> f(c(c(x1))) 210.12/60.04 , c(c(x1)) -> c(x1) } 210.12/60.04 Weak Trs: { f(x1) -> n(c(c(x1))) } 210.12/60.04 Obligation: 210.12/60.04 derivational complexity 210.12/60.04 Answer: 210.12/60.04 MAYBE 210.12/60.04 210.12/60.04 The weightgap principle applies (using the following nonconstant 210.12/60.04 growth matrix-interpretation) 210.12/60.04 210.12/60.04 TcT has computed the following triangular matrix interpretation. 210.12/60.04 Note that the diagonal of the component-wise maxima of 210.12/60.04 interpretation-entries contains no more than 1 non-zero entries. 210.12/60.04 210.12/60.04 [f](x1) = [1] x1 + [2] 210.12/60.04 210.12/60.04 [n](x1) = [1] x1 + [0] 210.12/60.04 210.12/60.04 [c](x1) = [1] x1 + [1] 210.12/60.04 210.12/60.04 [s](x1) = [1] x1 + [0] 210.12/60.04 210.12/60.04 The order satisfies the following ordering constraints: 210.12/60.04 210.12/60.04 [f(x1)] = [1] x1 + [2] 210.12/60.04 >= [1] x1 + [2] 210.12/60.04 = [n(c(c(x1)))] 210.12/60.04 210.12/60.04 [n(f(x1))] = [1] x1 + [2] 210.12/60.04 >= [1] x1 + [2] 210.12/60.04 = [f(n(x1))] 210.12/60.04 210.12/60.04 [n(s(x1))] = [1] x1 + [0] 210.12/60.04 ? [1] x1 + [2] 210.12/60.04 = [f(s(s(x1)))] 210.12/60.04 210.12/60.04 [c(f(x1))] = [1] x1 + [3] 210.12/60.04 ? [1] x1 + [4] 210.12/60.04 = [f(c(c(x1)))] 210.12/60.04 210.12/60.04 [c(c(x1))] = [1] x1 + [2] 210.12/60.04 > [1] x1 + [1] 210.12/60.04 = [c(x1)] 210.12/60.04 210.12/60.04 210.12/60.04 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 210.12/60.04 210.12/60.04 We are left with following problem, upon which TcT provides the 210.12/60.04 certificate MAYBE. 210.12/60.04 210.12/60.04 Strict Trs: 210.12/60.04 { n(f(x1)) -> f(n(x1)) 210.12/60.04 , n(s(x1)) -> f(s(s(x1))) 210.12/60.04 , c(f(x1)) -> f(c(c(x1))) } 210.12/60.04 Weak Trs: 210.12/60.04 { f(x1) -> n(c(c(x1))) 210.12/60.04 , c(c(x1)) -> c(x1) } 210.12/60.04 Obligation: 210.12/60.04 derivational complexity 210.12/60.04 Answer: 210.12/60.04 MAYBE 210.12/60.04 210.12/60.04 We use the processor 'matrix interpretation of dimension 2' to 210.12/60.04 orient following rules strictly. 210.12/60.04 210.12/60.04 Trs: { n(s(x1)) -> f(s(s(x1))) } 210.12/60.04 210.12/60.04 The induced complexity on above rules (modulo remaining rules) is 210.12/60.04 YES(?,O(n^2)) . These rules are moved into the corresponding weak 210.12/60.04 component(s). 210.12/60.04 210.12/60.04 Sub-proof: 210.12/60.04 ---------- 210.12/60.04 TcT has computed the following triangular matrix interpretation. 210.12/60.04 210.12/60.04 [f](x1) = [1 0] x1 + [0] 210.12/60.04 [0 1] [0] 210.12/60.04 210.12/60.04 [n](x1) = [1 1] x1 + [0] 210.12/60.04 [0 1] [0] 210.12/60.04 210.12/60.04 [c](x1) = [1 0] x1 + [0] 210.12/60.04 [0 0] [0] 210.12/60.04 210.12/60.04 [s](x1) = [1 0] x1 + [0] 210.12/60.04 [0 0] [1] 210.12/60.04 210.12/60.04 The order satisfies the following ordering constraints: 210.12/60.04 210.12/60.04 [f(x1)] = [1 0] x1 + [0] 210.12/60.04 [0 1] [0] 210.12/60.04 >= [1 0] x1 + [0] 210.12/60.04 [0 0] [0] 210.12/60.04 = [n(c(c(x1)))] 210.12/60.04 210.12/60.04 [n(f(x1))] = [1 1] x1 + [0] 210.12/60.04 [0 1] [0] 210.12/60.04 >= [1 1] x1 + [0] 210.12/60.04 [0 1] [0] 210.12/60.05 = [f(n(x1))] 210.12/60.05 210.12/60.05 [n(s(x1))] = [1 0] x1 + [1] 210.12/60.05 [0 0] [1] 210.12/60.05 > [1 0] x1 + [0] 210.12/60.05 [0 0] [1] 210.12/60.05 = [f(s(s(x1)))] 210.12/60.05 210.12/60.05 [c(f(x1))] = [1 0] x1 + [0] 210.12/60.05 [0 0] [0] 210.12/60.05 >= [1 0] x1 + [0] 210.12/60.05 [0 0] [0] 210.12/60.05 = [f(c(c(x1)))] 210.12/60.05 210.12/60.05 [c(c(x1))] = [1 0] x1 + [0] 210.12/60.05 [0 0] [0] 210.12/60.05 >= [1 0] x1 + [0] 210.12/60.05 [0 0] [0] 210.12/60.05 = [c(x1)] 210.12/60.05 210.12/60.05 210.12/60.05 We return to the main proof. 210.12/60.05 210.12/60.05 We are left with following problem, upon which TcT provides the 210.12/60.05 certificate MAYBE. 210.12/60.05 210.12/60.05 Strict Trs: 210.12/60.05 { n(f(x1)) -> f(n(x1)) 210.12/60.05 , c(f(x1)) -> f(c(c(x1))) } 210.12/60.05 Weak Trs: 210.12/60.05 { f(x1) -> n(c(c(x1))) 210.12/60.05 , n(s(x1)) -> f(s(s(x1))) 210.12/60.05 , c(c(x1)) -> c(x1) } 210.12/60.05 Obligation: 210.12/60.05 derivational complexity 210.12/60.05 Answer: 210.12/60.05 MAYBE 210.12/60.05 210.12/60.05 We use the processor 'matrix interpretation of dimension 4' to 210.12/60.05 orient following rules strictly. 210.12/60.05 210.12/60.05 Trs: { n(f(x1)) -> f(n(x1)) } 210.12/60.05 210.12/60.05 The induced complexity on above rules (modulo remaining rules) is 210.12/60.05 YES(?,O(n^4)) . These rules are moved into the corresponding weak 210.12/60.05 component(s). 210.12/60.05 210.12/60.05 Sub-proof: 210.12/60.05 ---------- 210.12/60.05 TcT has computed the following triangular matrix interpretation. 210.12/60.05 210.12/60.05 [1 0 1 0] [0] 210.12/60.05 [f](x1) = [0 1 0 1] x1 + [0] 210.12/60.05 [0 0 1 0] [1] 210.12/60.05 [0 0 0 1] [0] 210.12/60.05 210.12/60.05 [1 1 1 0] [0] 210.12/60.05 [n](x1) = [0 1 0 1] x1 + [0] 210.12/60.05 [0 0 1 1] [0] 210.12/60.05 [0 0 0 1] [0] 210.12/60.05 210.12/60.05 [1 0 0 0] [0] 210.12/60.05 [c](x1) = [0 0 0 0] x1 + [0] 210.12/60.05 [0 0 1 0] [0] 210.12/60.05 [0 0 0 0] [0] 210.12/60.05 210.12/60.05 [1 0 0 0] [0] 210.12/60.05 [s](x1) = [0 0 0 0] x1 + [0] 210.12/60.05 [0 0 0 0] [1] 210.12/60.05 [0 0 0 0] [1] 210.12/60.05 210.12/60.05 The order satisfies the following ordering constraints: 210.12/60.05 210.12/60.05 [f(x1)] = [1 0 1 0] [0] 210.12/60.05 [0 1 0 1] x1 + [0] 210.12/60.05 [0 0 1 0] [1] 210.12/60.05 [0 0 0 1] [0] 210.12/60.05 >= [1 0 1 0] [0] 210.12/60.05 [0 0 0 0] x1 + [0] 210.12/60.05 [0 0 1 0] [0] 210.12/60.05 [0 0 0 0] [0] 210.12/60.05 = [n(c(c(x1)))] 210.12/60.05 210.12/60.05 [n(f(x1))] = [1 1 2 1] [1] 210.12/60.05 [0 1 0 2] x1 + [0] 210.12/60.05 [0 0 1 1] [1] 210.12/60.05 [0 0 0 1] [0] 210.12/60.05 > [1 1 2 1] [0] 210.12/60.05 [0 1 0 2] x1 + [0] 210.12/60.05 [0 0 1 1] [1] 210.12/60.05 [0 0 0 1] [0] 210.12/60.05 = [f(n(x1))] 210.12/60.05 210.12/60.05 [n(s(x1))] = [1 0 0 0] [1] 210.12/60.05 [0 0 0 0] x1 + [1] 210.12/60.05 [0 0 0 0] [2] 210.12/60.05 [0 0 0 0] [1] 210.12/60.05 >= [1 0 0 0] [1] 210.12/60.05 [0 0 0 0] x1 + [1] 210.12/60.05 [0 0 0 0] [2] 210.12/60.05 [0 0 0 0] [1] 210.12/60.05 = [f(s(s(x1)))] 210.12/60.05 210.12/60.05 [c(f(x1))] = [1 0 1 0] [0] 210.12/60.05 [0 0 0 0] x1 + [0] 210.12/60.05 [0 0 1 0] [1] 210.12/60.05 [0 0 0 0] [0] 210.12/60.05 >= [1 0 1 0] [0] 210.12/60.05 [0 0 0 0] x1 + [0] 210.12/60.05 [0 0 1 0] [1] 210.12/60.05 [0 0 0 0] [0] 210.12/60.05 = [f(c(c(x1)))] 210.12/60.05 210.12/60.05 [c(c(x1))] = [1 0 0 0] [0] 210.12/60.05 [0 0 0 0] x1 + [0] 210.12/60.05 [0 0 1 0] [0] 210.12/60.05 [0 0 0 0] [0] 210.12/60.05 >= [1 0 0 0] [0] 210.12/60.05 [0 0 0 0] x1 + [0] 210.12/60.05 [0 0 1 0] [0] 210.12/60.05 [0 0 0 0] [0] 210.12/60.05 = [c(x1)] 210.12/60.05 210.12/60.05 210.12/60.05 We return to the main proof. 210.12/60.05 210.12/60.05 We are left with following problem, upon which TcT provides the 210.12/60.05 certificate MAYBE. 210.12/60.05 210.12/60.05 Strict Trs: { c(f(x1)) -> f(c(c(x1))) } 210.12/60.05 Weak Trs: 210.12/60.05 { f(x1) -> n(c(c(x1))) 210.12/60.05 , n(f(x1)) -> f(n(x1)) 210.12/60.05 , n(s(x1)) -> f(s(s(x1))) 210.12/60.05 , c(c(x1)) -> c(x1) } 210.12/60.05 Obligation: 210.12/60.05 derivational complexity 210.12/60.05 Answer: 210.12/60.05 MAYBE 210.12/60.05 210.12/60.05 None of the processors succeeded. 210.12/60.05 210.12/60.05 Details of failed attempt(s): 210.12/60.05 ----------------------------- 210.12/60.05 1) 'empty' failed due to the following reason: 210.12/60.05 210.12/60.05 Empty strict component of the problem is NOT empty. 210.12/60.05 210.12/60.05 2) 'Fastest' failed due to the following reason: 210.12/60.05 210.12/60.05 None of the processors succeeded. 210.12/60.05 210.12/60.05 Details of failed attempt(s): 210.12/60.05 ----------------------------- 210.12/60.05 1) 'Fastest' failed due to the following reason: 210.12/60.05 210.12/60.05 None of the processors succeeded. 210.12/60.05 210.12/60.05 Details of failed attempt(s): 210.12/60.05 ----------------------------- 210.12/60.05 1) 'matrix interpretation of dimension 6' failed due to the 210.12/60.05 following reason: 210.12/60.05 210.12/60.05 The input cannot be shown compatible 210.12/60.05 210.12/60.05 2) 'matrix interpretation of dimension 5' failed due to the 210.12/60.05 following reason: 210.12/60.05 210.12/60.05 The input cannot be shown compatible 210.12/60.05 210.12/60.05 3) 'matrix interpretation of dimension 4' failed due to the 210.12/60.05 following reason: 210.12/60.05 210.12/60.05 The input cannot be shown compatible 210.12/60.05 210.12/60.05 4) 'matrix interpretation of dimension 3' failed due to the 210.12/60.05 following reason: 210.12/60.05 210.12/60.05 The input cannot be shown compatible 210.12/60.05 210.12/60.05 5) 'matrix interpretation of dimension 2' failed due to the 210.12/60.05 following reason: 210.12/60.05 210.12/60.05 The input cannot be shown compatible 210.12/60.05 210.12/60.05 6) 'matrix interpretation of dimension 1' failed due to the 210.12/60.05 following reason: 210.12/60.05 210.12/60.05 The input cannot be shown compatible 210.12/60.05 210.12/60.05 210.12/60.05 2) 'Fastest (timeout of 30 seconds)' failed due to the following 210.12/60.05 reason: 210.12/60.05 210.12/60.05 Computation stopped due to timeout after 30.0 seconds. 210.12/60.05 210.12/60.05 3) 'iteProgress' failed due to the following reason: 210.12/60.05 210.12/60.05 Fail 210.12/60.05 210.12/60.05 4) 'bsearch-matrix' failed due to the following reason: 210.12/60.05 210.12/60.05 The input cannot be shown compatible 210.12/60.05 210.12/60.05 210.12/60.05 210.12/60.05 3) 'iteProgress (timeout of 297 seconds)' failed due to the 210.12/60.05 following reason: 210.12/60.05 210.12/60.05 Fail 210.12/60.05 210.12/60.05 4) 'bsearch-matrix (timeout of 297 seconds)' failed due to the 210.12/60.05 following reason: 210.12/60.05 210.12/60.05 The input cannot be shown compatible 210.12/60.05 210.12/60.05 210.12/60.05 Arrrr.. 210.12/60.06 EOF